On Free Baxter Algebras: Completions and the Internal Construction

Advances in Mathematics 151, 101�127 (2000)

On Free Baxter Algebras: Completions and theInternal Construction1

Li Guo and William Keigher

Department of Mathematics and Computer Science, Rutgers University,Newark, New Jersey 07102

E-mail: liguo�newark.rutgers.edu, keigher�newark.rutgers.edu

Received January 13, 1999; accepted March 11, 1999

1. INTRODUCTION

In a previous paper [4], we gave an explicit construction of a free Baxteralgebra. This construction is called the shuffle Baxter algebra since it isdescribed in terms of mixable shuffles. In this paper and its sequel [5], wewill continue the study of free Baxter algebras.

There are two goals of this paper. The first goal is to extend the con-struction of shuffle Baxter algebras to completions of Baxter algebras. Thisprocess is motivated by a construction of Cartier [2] and is analogous tothe process of completing a polynomial algebra to obtain a power seriesalgebra. However, as we will see later, unlike the close similarity of proper-ties of a polynomial algebra and a power series algebra, properties of ashuffle Baxter algebra and its completion can be quite different.

The second goal is to establish a connection between the shuffle Baxteralgebra we have constructed to the standard Baxter algebra constructed byRota [10]. The shuffle Baxter algebra is an external construction in thesense that it is a free Baxter algebra obtained without reference to anyother Baxter algebra. On the other hand, the standard Baxter algebra is aninternal construction, obtained as a Baxter subalgebra inside a naturallydefined Baxter algebra constructed originally by Baxter [1]. There areseveral restrictions on Rota's original construction of standard Baxteralgebras. By modifying Rota's method and making use of the shuffle Baxteralgebra construction, we are able to construct the standard Baxter algebrain full generality. The shuffle product construction of a free Baxter algebrahas the advantage that its module structure and Baxter operator can beeasily described. The description of a free Baxter algebra as a standardBaxter algebra has the advantage that its multiplication is very simple. We

doi:10.1006�aima.1999.1867, available online at http:��www.idealibrary.com on

1010001-8708�00 �35.00

Copyright � 2000 by Academic PressAll rights of reproduction in any form reserved.

1 The first author is supported in part by NSF Grant DMS 97-96122.

will give an explicit description of the isomorphism between the shuffleBaxter algebra and the standard Baxter algebra. This description willenable us to make use of properties of both the shuffle product descriptionand Rota's description of free Baxter algebras. Some applications will begiven in [5].

We will start with a brief summary of definitions and basic properties ofthe shuffle Baxter algebra in Section 2. We also take the opportunity toextend the construction of shuffle Baxter algebras to the category of Baxteralgebras not necessarily having an identity. In Section 3, we define the com-pletion of a Baxter algebra by making use of the filtration given by theBaxter operator and give a description of a free complete Baxter algebra interms of mixable shuffles. In Section 4, we construct the standard Baxteralgebras, generalizing Rota. Variations of the construction for completeBaxter algebras and for Baxter algebras not necessarily having an identityare also considered.

2. SHUFFLE BAXTER ALGEBRAS

We write N for the additive monoid of natural numbers [0, 1, 2, ...] andN+=[n # N | n>0] for the positive integers.

Let Rings denote the category of commutative rings with identity. Forany C # Rings, let AlgC denote the category of C-algebras with identity.For C # Rings and for any C-modules M and N, the tensor product M�Nis taken over C unless otherwise indicated. Let M be a C-module. Forn # N, denote

M�n=M� } } } �M

n factors

with the convention that M�0=C.

2.1. Baxter Algebras

Baxter algebras were first studied by Baxter [1] and the category ofBaxter algebras was first studied by Rota [10]. We recall basic definitionsand properties of Baxter algebras. See [4, 10] for details.

Definition 2.1. Let C be a ring, * # C, and let R be a C-algebra.

v A Baxter operator of weight * on R over C is a C-moduleendomorphism P of R satisfying

P(x) P( y)=P(xP( y))+P( yP(x))+*P(xy), x, y # R. (1)

102 GUO AND KEIGHER

v A Baxter C-algebra of weight * is a pair (R, P) where R is aC-algebra and P is a Baxter operator of weight * on R over C.

v Let (R, P) and (S, Q) be two Baxter C-algebras of weight *.A homomorphism of Baxter C-algebras f : (R, P) � (S, Q) is a homomorphismf : R � S of C-algebras with the property that f (P(x))=Q( f (x)) for allx # R.

If the meaning of * is clear, we will suppress * from the notation. Notethat our * is &q in the notation of Rota [11].

Let BaxC, * denote the category of Baxter C-algebras of weight *.A Baxter ideal of (R, P) is an ideal I of R such that P(I)�I. Other con-cepts of C-algebras, such as subalgebra and quotient algebra, can also bedefined for Baxter algebras [4].

2.2. Shuffle Baxter Algebras with an Identity

Let A # AlgC . In a previous work [4], we used mixable shuffles to con-struct a mixable shuffle algebra �C (A) and proved that it is a free BaxterC-algebra on A.

For m, n # N+ , define the set of (m, n)-shuffles by

S(m, n)={_ # Sm+n } _&1(1)<_&1(2)< } } } <_&1(m),_&1(m+1)<_&1(m+2)< } } } <_&1(m+n)= .

Given an (m, n)-shuffle _ # S(m, n), a pair of indices (k, k+1), 1�k<m+n is called an admissible pair for _ if _(k)�m<_(k+1). Denote T_

for the set of admissible pairs for _. For a subset T of T_, call the pair(_, T ) a mixable (m, n)-shuffle, where (_, T ) is identified with _ if T is theempty set. Denote

S� (m, n)=[(_, T ) | _ # S(m, n), T/T_]

for the set of (m, n)-mixable shuffles.For m, n # N+ , denote x=x1 � } } } �xm # A�m, and y= y1 � } } } �

yn # A�n. For _ # Sm , denote

_(x)=x_(1) �x_(2) � } } } �x_(m) .

Denote x�y=x1 � } } } �xm �y1 � } } } �yn # A� (m+n), and, for _ # Sm+n ,denote

_(x�y)=u_(1) �u_(2) � } } } �u_(m+n) ,

103ON FREE BAXTER ALGEBRAS

where

uk={ xk ,yk&m ,

1�k�m.m+1�k�m+n.

Definition 2.2. Let x # A�m, y # A�n, and _ # S(m, n).

(1) _(x�y) # A� (m+n) is called a shuffle of x and y.

(2) Let T be a subset of T_ . The element

_(x�y; T )=u_(1) �� u_(2) �� } } } �� u_(m+n) ,

where for each pair (k, k+1), 1�k<m+n,

u_(k) �� u_(k+1)={u_(k) u_(k+1) ,u_(k) �u_(k+1) ,

(k, k+1) # T(k, k+1) � T

is called a mixable shuffle of x and y.

Fix a * # C. For x=x0 �x1 � } } } �xm # A� (m+1) and y=y0 �y1 �} } } �yn # A� (n+1), define

xhy= :(_, T ) # S� (m, n)

* |T |x0 y0 �_(x�y; T ) # �k�m+n+1

A�k.

Then h extends to a mapping

h : A� (m+1)_A� (n+1) � �k�m+n+1

A�k, m, n # N

by C-linearity. Let

�C (A)=�C (A, *)= �k # N

A� (k+1)=A�A�2� } } }

Extending by additivity, the map h gives a C-bilinear map

h : �C (A)_�C (A) � �C (A)

with the convention that

A_A� (m+1) � A� (m+1)

is the scalar multiplication on the left A-module A� (m+n). Define a C-linearendomorphism PA on �C (A) by assigning

PA(x0 �x1 � } } } �xn)=1A �x0 �x1 � } } } �xn ,

104 GUO AND KEIGHER

for all x0 �x1 � } } } �xn # A� (n+1) and extending by additivity. LetjA : A � �C (A) be the canonical inclusion map. We proved the followingtheorem in [4].

Theorem 2.3. (1) The C-module �C (A), together with the multiplicationh, is a commutative C-algebra with an identity.

(2) (�C (A), PA), together with the natural embedding jA : A ��C (A), is a free Baxter C-algebra on A (of weight *). In other words, forany Baxter C-algebra (R, P) and any C-algebra map . : A � R, there existsa unique Baxter C-algebra homomorphism .~ : (�C (A), PA) � (R, P) suchthat the diagram

commutes.

The Baxter C-algebra (�C (A), PA) will be called the shuffle BaxterC-algebra (of weight *) on A. When there is no danger of confusion, we willoften suppress the symbol h and simply denote xy for xhy in �C (A).

For a given set X, let C[X ] be the polynomial C-algebra on X with thenatural embedding X /�C[X ]. Let (�C (X ), PX) be the Baxter C-algebra(�C (C[X ]), PC[X ]). (�C (X ), PX) will be called the shuffle BaxterC-algebra (of weight *) on X.

Proposition 2.4. (�C (X ), PX), together with the set embedding

jX : X/�C[X ] ww�jC [X ]

�C (C[X ]),

is a free Baxter C-algebra on the set X, described by the following universalproperty: For any Baxter C-algebra (R, P) over C and any set map. : X � R, there exists a unique Baxter C-algebra homomorphism.~ : (�C (X ), PX) � (R, P) such that the diagram

commutes.


If we choose A=C in the construction of free Baxter C-algebras, thenwe get

�C (C )= ��

n=0

C� (n+1)= ��

n=0

C1� (n+1),

where

1� (n+1)=1C � } } } �1C

(n+1)-factors

.

Thus �C (C ) is a free C-module on the basis 1�n, n�1.

Proposition 2.5. For any m, n # N,

1� (m+1)h1� (n+1)= :m

k=0\m+n&k

n +\nk+ *k1� (m+n+1&k).

2.3. Shuffle Baxter Algebras without an Identity

We now construct a shuffle Baxter algebra in the category of Baxteralgebras not necessarily having an identity.

For C # Rings, let Alg0C be the category of C-algebras not necessarily

having an identity and let Bax0C be the category of Baxter C-algebras not

necessarily having an identity. For A # Alg0C , we will use mixable shuffles to

construct a free Baxter algebra on A in Bax0C . This construction was given

in a special case in [4]. A similar construction can be carried out if C isassumed to be a commutative ring not necessarily having an identity, butwe will not given details here.

Let C # Rings and A # Alg0C be given. We use a well-known construction

[2, 6] to embed A in an element A+ # AlgC . Let A+=C�A with theaddition defined componentwise and the multiplication defined by

(c, a)(d, b)=(cd, cb+da+ab), c, d # C, a, b # A.

Then A+ is in AlgC with (1C , 0) as the identity and a [ (0, a) embeds Ain A+ as a subobject in Alg0

C (in fact, as an ideal).Define

�C (A)0= �n # N

((A+)�n�A)

with the convention that (A+)�0=C. Thus �C (A)0 is the C-submodule of�C (A+) generated by tensors of the form

x0 � } } } �xn , xi # A+, 0�i�n&1, xn # A.

106 GUO AND KEIGHER

Since any mixable shuffle of x0 � } } } �xm and y0 � } } } �yn has eitherxm or yn or xm yn as the last tensor factor, we see that �C (A)0 is aC-subalgebra of �C (A+). It is also clearly closed under the Baxteroperator PA+ . So �C (A)0, with the restriction of PA+ , denoted by PA , isa subobject of �C (A+) in Bax0

C . It is called the shuffle Baxter algebra onA (of weight *) in the category Bax0

C .

Proposition 2.6. (�C (A)0, PA), together with the natural embeddingjA : A � �C (A)0, is a free Baxter C-algebra on A (of weight *).

Proof. for any (R, P) # Bax0C and any morphism .: A � R in Alg0

C , wewill display a unique morphism .~ : (�C (A)0, PA) � (R, P) in Bax0

C thatextends .. For each n # N, we will define a C-linear map

.~ n : (A+)�n�A � R.

If n=0, we define

.~ 0 : (A+)�0�A=A � R

by .~ 0(x0)=.(x0), x0 # A. Assuming .~ n is defined, we define

.n+1 : (A+)n+1_A � R

by

.n+1(x0 , ..., xn+1)=cP(.~ n(x1 � } } } �xn+1))

+.(x$0) P(.~ n(x1 � } } } �xn+1)),

if x0=(c, x$0) # A+=C�A. Using the induction hypothesis, we see thatthis map is C-multilinear, and so induces

.~ n+1 : (A+)� (n+1) �A � R.

We then use .~ n , n # N to define

.~ = :�

n=0

.~ n : �C (A)0= ��

n=0

((A+)�n�A) � R. (2)

Since the products of �C (A)0��C (A+) and A both satisfy the mixableshuffle product identity (see [4, Proposition 4.2]), .~ is a morphismin Bax0

C . On the other hand, for x0 � } } } �xn # (A+)�n�A withx0=(c, x$0), we have

x0 � } } } �xn=cP1(x1 � } } } �xn)+.(x$0) PA(x1 � } } } �xn).


Thus Eq. (2) is the only possible way to define a morphism in Bax0C from

�C (A)0 to R that extends .. This verifies the required universal propertyof �C (A)0. K

3. COMPLETE BAXTER ALGEBRAS

We will define a natural decreasing filtration on Baxter algebras andstudy the associated completion. We will show that the completion of ashuffle Baxter algebra is a free object in the category of complete Baxteralgebras. In this section, we retain the assumption that all algebras have anidentity.

3.1. Filtrations and Completions

Let (R, P) be a Baxter algebra. For any subset U of R, denote (U )B forthe Baxter ideal of R generated by U. We will define a decreasing filtrationFiln R of ideals of (R, P) as follows. Define Fil0 R=R. For any n # N,assume that Filn R is defined, and inductively define

Filn+1 R=(P(Filn R)) B .

Thus, for example,

Fil1 R=(P(R)) B and Fil2 R=(P((P(R)) B)) B .

Since each Filn R is a Baxter ideal of R, we have P(Filn R)�Filn R.Therefore

Filn+1R=(P(Filn R)) B �Filn R.

So [Filn R]n # N defines a decreasing filtration of Baxter ideals on (R, P).Assuming Fil1 R{R, then each R�Filn R, n # N+ , is a Baxter C-algebra.Since each of the projections R�Filn+1 R � R�Filn R, n # N+ is a BaxterC-algebra homomorphism, the inverse limit �(R�Filn R) is also a BaxterC-algebra.

Definition 3.1. Let (R, P) be a Baxter algebra.

(1) The decreasing filtration Filn R on R is called the Baxter filtrationon R.

(2) The Baxter algebra (R, P) is called proper if Fil1 R is a properBaxter ideal of R.

(3) Denote Bax$C for the subcategory of BaxC consisting of properBaxter C-algebras.

108 GUO AND KEIGHER

(4) For (R, P) # Bax$C , the inverse limit R� =�(R�Filn R) with theinduced Baxter operator P� is called the (Baxter) completion of (R, P).

(5) (R, P) # Bax$C is called (Baxter) complete if the natural BaxterC-algebra homomorphism

?R : R � �(R�Filn R)

is an isomorphism.

Let * # C and A # AlgC . It is easy to see (Proposition 3.5) that the shuffleBaxter algebra �C (A) of weight * is proper. On the other hand, for * # C,define P* on A by P*(a)=&*a. Then (A, P*) is a Baxter algebra of weight*. If * is invertible in A, then P*(A)=&*A=A. So (A, P*) is not proper.If * is not invertible in A, then (A, P*) is proper. In fact, the Baxtercompletion of (A, P*) is the same as � A�*nA, the *-adic completion of A.

Proposition 3.2. For each f: (R, P) � (S, Q) in Bax$C , there is a uniquef� : (R� , P� ) � (S� , Q� ) in BaxC making the diagram

f

f�

R S?R ?S

R� S�

commute.

Before proving the proposition, we first give an elementary fact onBaxter algebras.

Lemma 3.3. Let f : (R, P) � (S, Q) be a morphism in Bax$C . For anysubset U of R, we have f ((U ) B)�( f (U )) B .

Proof. For a given subset U of R, consider the morphism

R w�f S � S�( f (U )) B

in BaxC . Since U is in the kernel of the morphism, it follows that (U ) B

is in the kernel of the morphism. Thus f ((U )B)�( f (U )) B . K

Proof of Proposition 3.2. We first apply induction on k to show thatf (Filk R)/Filk S. For k=0, this just says f (R)�S. Assume thatf (Filk R)�Filk R. Applying Lemma 3.3 to P(Filk R), we obtain


f (Filk+1 R)=f ((P(Filk R)) B)

�( f (P(Filk R))) B

=(Q( f (Filk R))) B

�(Q(Filk S)) B

=Filk+1 S.

This completes the induction. Then the proposition follows from generalresults on completions [9, p. 57]. K

3.2. An Alternative Description

We now give an interpretation of a complete free Baxter algebra in termsinfinite sequences. We consider the following situation. Let R be aC-algebra and let Ri , i # N, be C-submodules of R such that

(1) R=�k # N Rk as a C-module, and

(2) F nR= def �k>n Rk is an ideal of R, n # N.

We will define a multiplication on >k # N Rk . The definition is similar tothe case when R is a graded C-algebra. For lack of a suitable reference, wegive details below.

Fix a k # N. Let (x(n))n be a sequence of elements in Rk . If there is ann0 # N such that x(n)=x(n0) for n�n0 , then define limn � � x (n)=x(n0) # Rk .Further, let (x(n))=((x (n)

k )k) be a sequence of elements in >k # N Rk . Iflimn � � x (n)

k exists for each k, then define

limn � �

x(n)=( limn � �

x (n)k )k # `

k

Rk .

For any x=(xk)k # >k Rk and any n # N, define x[n]=(x[n]k )k # >k # N Rk

by

x[n]k ={xk ,

0,k�n,k>n.

Now let x=(xk)k and y=( yk)k be two elements of >k # N Rk . For givenn # N, we have x[n], y[n] # R=�k Rk . So x[n]y[n] can be uniquelyexpressed as (z (n)

k )k , z (n)k # Rk and z (n)

k =0 for k>>0. For each fixed k, weobtain a sequence (z (n)

k )n in Rk . When n�k, we have x[n]=x[k]+x$ andy[n]= y[k]+ y$ with x$, y$ # F k+1R. Since F k+1R is an ideal of R, we have

x[n]y[n]#x[k]y[k] (mod F k+1R).

110 GUO AND KEIGHER

Therefore z(n)k =z(k)

k for n�k and limn � � z (n)k # Rk is well defined. Define

(xk)k ( yk)k=( limn � �

z (n)k )k # `

k

Rk .

In other words,

(xk)k ( yk)k= limn � �

x[n]y[n].

It can be easily verified that this defines an associative, commutativemultiplication on >k # N Rk , making it into a commutative C-algebra.

Proposition 3.4. Let R be a C-algebra and let Ri , i # N, be C-submodulesof R such that

(1) R=�k # N Rk as a C-module, and(2) F nR =

def�k>n Rk is an ideal of R, n # N.

There is a unique C-algebra isomorphism

�R : �(R�F kR)$ `k # N

Rk

that makes the diagram in AlgC

�R

R `k # N

Rk

?R

�(R�F kR)

commute.

Proof. By definition, �(R�F kR) is the inverse limit of the inversesystem pn+1, n : R�F n+1R � R�F nR, n�1. An element ((x (n)

k )k+F nR)n #>n # N R�F nR is an element of �(R�F kR) if and only if, for any n�1,

pn+1, n((x (n+1)k )k+F n+1R)=(x (n)

k )k+F nR.

Since R�F nR$�k�n Rk , this is so if and only if x (n+1)k =x (n)

k for k�n.Therefore ((x (n)

k ) k+F nR)n is in �(R�F kR) if and only if there is( yn)n # >n # N Rn such that, for any n, x (n)

k = yk for k�n. In fact, we cantake yk=x (k)

k . This gives the desired map �R : �(R�F kR) � >k # N Rk .More precisely, we have

�R(((x (n)k )k+F nR)n)=(x (k)

k )k . (3)


For x=(xk)k # R=�k # N Rk , ?R(x) # �(R�F kR) is defined to be thesequence ((xk)k+F nR)n which corresponds under �R to the element(xk)k # >k # N Rk . This proves the commutativity of the diagram. K

3.3. Complete Shuffle Baxter Algebras

We now consider the completion of �C(A). Recall that we denote�k

C (A) for the C-submodule A � (k+1) of �C(A). We denote � �C(A)�Filk �C(A) by �� C(A).

Proposition 3.5. Given k # N+ ,

(1) Filk �C(A)=�n�k �nC (A),

(2) Filk �C(A) is a Baxter homogeneous ideal of �C(A), and(3) The quotient Baxter C-algebra �C(A)�Filk �C(A) is isomorphic

to �k&1n=0 �n

C (A) as a C-module.

Proof. (1) It follows from the definition of h that, for any C-algebra A,

�mC (A)h�n

C (A)� :m+n

k=max[m, n]

�kC (A).

This shows that �n�k �nC (A) is a Baxter ideal. Next we prove

Filk �C(A)= �n�k

�nC (A) (4)

by induction on k. By definition, Fil1 �C(A) is the Baxter ideal generatedby PA(�C(A))=1A ��C(A). On the other hand. �n�1 �n

C (A) equalsAh (1A ��C(A)), hence is also generated by 1A ��C(A). This verifiesEq. (4) for k=1. Assume that Filk �C(A)=�n�k �n

C (A) for a k # N+.From this we obtain that Filk+1 �C(A) is the Baxter ideal generated by

PA(Filk �C(A))=PA \�n�k

�nC (A)+=1A �\�

n�k

�nC (A)+ .

On the other hand, �n�k+1 �nC (A) equals Ah (1A ��n�k �n

C (A)),hence is also generated by 1A ��n�k �n

C (A). This verifies Eq. (4) fork+1.

Other statements in the proposition follow immediately from the firstone. K

It follows from Proposition 3.5 that R=�C(A)=�k # N �kC (A) satisfies

the two conditions for R in Proposition 3.4. Thus the product >k # N

�kC (A) is a C-algebra. Define an operator P� on this product by P� ((xk))=

(1A �xk&1) with the convention that 1A �xk&1=0 for k=0.

112 GUO AND KEIGHER

Theorem 3.6. (1) P� is a Baxter operator on >k # N �kC (A), and

�A =def ��C (A) : �� C(A) � >k # N �k

C (A) is an isomorphism of BaxterC-algebras.

(2) Given a morphism f : A � B in AlgC , we have the following com-mutative diagram in BaxC

�B

�A�� C (A) `

k # N

�kC (A)

�� C ( f ) >k fk

�� C (B) `k # N

�kC (B)

where �� C( f ) is induced by �C( f ) which is in turn induced by f, and fk :�k

C (A) � �kC (B) is the tensor power morphism of C-modules f � (k+1):

A� (k+1) � B� (k+1) induced from f.

Proof. (1) Let ((x (n)k )k+Filn �C(A))n # �� C(A) be given. Using

formula (3) for the map �A , we have

(P� b �A)(((x (n)k )k+Filn �C(A))n)=P� ((x (k)

k )k)

=(1A �x (k)k&1)k ,

and

(�A b P� )(((x (n)k )k+Filn �C(A))n)

=�A(((1A �x (n)k&1)k+Filn �C(A))n)

=(1A �x (k)k&1)k .

Thus P� b �A=�A b P� . Since �A is an isomorphism of C-algebras and sinceP� is known to satisfy the identity defining a Baxter operator, the aboveequation implies that the same identity is satisfied by P� .

(2) Given ((x (n)k )k+Filn �C(A))n # �� C(A), using formula (3) we

have

\`k

fk b �A+ (((x (n)k)k+Filn �C(A))n)=`k

fk((x (k)k )k)

=( fk(x (k)k ))k


and

(�A b �� C(A))(((x (n)k )k+Filn �C(A))n)

=�A((( fk(x (n)k ))k+Filn �C(B))n)

=( fk(x (k)k ))k .

This proves that the diagram commutes. K

As an example of Theorem 3.6, consider the case when A=C and *=0.Let HC be the ring of Hurwitz series over C [7], defined to be the set ofsequences

[(an) | an # C, n # N]

in which the addition is defined componentwise and the multiplication isdefined by

(an)(bn)=(cn)

with

cn= :n

k=0 \nk+ ak bn&k .

Denote en for the sequence (ak) in which an=1C and ak=0 for k{n. Sinceenem=( m+n

n ) em+n , the following corollary follows from Proposition 2.5.

Corollary 3.7. The assignment

1� (n+1) [ en , n�0

defines an isomorphism

�� C(C ) � HC.

By Theorem 3.6 and part three of Proposition 3.5, we have theisomorphism of inverse systems

�� C(A)�Filk �� C(A)$�(A)�Filk �C(A).

Thus the completion of �� C(A) is itself, so it is complete. We next verify the

free universal property of �� C(A) in the category BaxC@ of complete BaxterC-algebras. Abbreviate ?A for ?�C (A) .

114 GUO AND KEIGHER

Theorem 3.8. (�� C (A), P� A), together with the natural embeddingjA : A w�

jA�C (A) w�

?A�� C (A), is a free complete Baxter C-algebra on A (of

weight *). In other words, for any complete Baxter C-algebra (R, P) andany C-algebra map . : A � R, there exists a unique Baxter C-algebrahomomorphism . : (�� C (A), P� A) � (R, P) such that the diagram

commutes.

Proof. Given a (R, P) and . : A � R as in the statement of the theorem,by the universal property of (�C (A), jA) in BaxC , there is a unique.~ : �C (A) � R in BaxC such that .~ b jA=.. Since R is complete, byProposition 3.2, there is

. =def .~^ : �� C (A) � R� $R

such that . b ?A=.~ . Then we have

. b jA=. b ?A b jA=.~ b jA=..

This proves the existence of .. The uniqueness follows from theuniqueness of .~ and the uniqueness of the completion. K

3.4. Completions of Baxter Algebras

It is clear that complete Baxter C-algebras, together with the Baxter

algebra homomorphisms between them, form a full subcategory BaxC@ ofBaxC . Recall that we denote Bax$C for the full subcategory of BaxC consist-

ing of proper Baxter algebras. Denoting IC : BaxC@ � Bax$C for the naturalinclusion of categories, we then have

Proposition 3.9. (1) For any R # Bax$C , the Baxter completion R� of Ris complete.

(2) The assignments (R, P) [ (R� , P� ) and f [ f� define a functor FC

from Bax$C to BaxC@ , and the morphisms ?R : R � R� , R # Bax$C , define anatural transformation between the identity functor on Bax$C and the functorIC b FC : Bax$C � Bax$C .


Proof. We only need to prove that R� is complete. The rest of the proofis clear.

Note that the filtration Filk on R� is not defined to be the naturalfiltration induced by the filtration Filk on R. So the general results oncompletions do not apply. Instead, we will use the facts that any Baxteralgebra is a quotient of a shuffle Baxter algebra, and that the completionof a shuffle Baxter algebra is complete, which follows from Proposition 3.5and Theorem 3.6.

For a given (R, P) # Bax$C , by the universal property of free Baxteralgebras, there is an A # AlgC and a surjective morphism � : �C (A) � R inBaxC . We will prove by induction on n # N+ that

�(Filn �C (A))=Filn R. (5)

For n=1 we have

�(Fil1 �C (A))=�(�C (A) PA(�C (A)))

=�(�C (A)) P(�(�C (A)))

=RP(R).

Note that RP(R) is the ideal of R generated by P(R). Since P(RP(R))�P(R), it is in fact the Baxter ideal of R generated by P(R). Thus RP(R)=Fil1 R. So Eq. (5) holds for n=1.

Assume that the equation holds for n. Part one of Proposition 3.5 showsthat Filn+1 �C (A) is the ideal of �C (A) generated by PA(Filn �C (A)).Then by induction we have

�(Filn+1 �C (A))=�(�C (A) PA(Filn �C (A)))

=�(�C (A)) P(�(�C (A)))

=RP(Filn R).

Since

P(RP(Filn R))=P(�(�C (A)) P(�(Filn �C (A))))

=�(PA(�C (A) PA(Filn �C (A))))

=�(PA(Filn+1 �C (A)))

��(�C (A) PA(Filn �C (A)))

=RP(Filn R),

RP(Filn R) is the Baxter ideal of R generated by P(Filn R), so is equal toFiln+1 R. This completes the induction.

116 GUO AND KEIGHER

Because of Eq. (5), the morphism � : �C (A) � R induces a morphism

�n : �C (A)�Filn �C (A) � R�Filn R

for each n # N+ and the kernel of �n is (ker �+Filn �C (A))�Filn �C (A)which is isomorphic to ker ��(ker , & Filn �C (A)). Thus we have the exactsequence of inverse systems

0 � (ker �+Filn �C (A))�Filn �C (A)

� �C (A)�Filn �C (A) � R�Filn R � 0

and the transition map of the left inverse system is identified with thenatural map

ker ��(ker � & Filn+1 �C (A)) � ker ��(ker � & Filn �C (A))

so is surjective. By [13, Lemma 3.5.3], for the first derived functor R1 �

of the inverse limit, we have

R1 �(ker �+Filn �C (A))�Filn �C (A)=0.

Therefore the above exact sequence of inverse systems gives the surjectivemorphism

�� : �� C (A) � R� .

Because of Theorem 3.6, Filn+1 �� C (A) is the ideal of �� C (A) generatedby P� A(Filn �� C (A)). Then the same argument for �: �C (A) � R in theprevious part of the proof can be repeated for the morphism �� : �� C (A) �R� . In particular, we have, for any n # N+ ,

�(Filn �� C (A))=Filn R� . (6)

We then obtain a surjective morphism

�� : �� C (A) � R�� .

Since �� C (A) is its own completion, we have the commutative diagram

?R�

�� C (A) t= �� C (A)

��

R� R��


in which both of the vertical maps are surjective. By the commutativity ofthe diagram, ?R� is surjective. Since �n Filn �� C (A)=0, by Eq. (6), we have�n Filn R� =0. Thus ?R� is injective. Therefore, R� is complete. K

4. THE STANDARD BAXTER ALGEBRA

The standard Baxter algebra constructed by Rota in [10] is a free objectin the category Bax0

C of Baxter algebras not necessarily having an identity.It is described as a Baxter subalgebra of another Baxter algebra whose con-struction goes back to Baxter [1]. In Rota's construction, there are furtherrestrictions that C is a field of characteristic zero, the free Baxter algebraobtained is on a finite set X, and the weight * is 1. By making use of shuffleBaxter algebras, we will show that Rota's description can be modified toyield a free Baxter algebra on an algebra in the category BaxC of Baxteralgebras with an identity, with a mild restriction on the weight *. We willalso provide a similar construction for algebras not necessarily having anidentity, and for complete Baxter algebras.

4.1. The Standard Baxter Algebra of Rota

We first briefly recall the construction of Rota of a standard Baxteralgebra S(X ) on a set X. For details, see [10, 12].

As before, let C be a commutative ring with an identity, and fix * # C.Let X be a given set. For each x # X, let t(x) be a sequence t(x)=(t (x)

1 , ..., t (x)n , ...) of distinct symbols t (x)

n . We also require that the sets[t (x1)

n ]n and [t (x2)n ]n are disjoint for x1 {x2 in X. Denote

X� = .x # X

[t (x)n | n # N+]

and denote A(X ) for the ring of sequences with entries in C[X� ], theC-algebra of polynomials with variables in X� . Thus the addition, multi-plication, and scalar multiplication by C[X� ] in A(X ) are definedcomponentwise. It will be useful to have the following description of A(X ).For k # N+ , denote #k for the sequence ($n, k)n , where $n, k is theKronecker delta. Then we can identify a sequence (an)n in A(X ) with aseries

:�

n=1

an#n=a1#1+a2#2+ } } } .

Then the addition, multiplication, and scalar multiplication by C[X� ] aregiven termwise.

118 GUO AND KEIGHER

Define

P$X=P$X, * : A(X ) � A(X )

by

P$X (a1 , a2 , a2 , ...)=*(0, a1 , a1+a2 , a1+a2+a3 , ...).

In other words, each entry of P$X (a), a=(a1 , a2 , ...), is * times the sum ofthe previous entries of a. If elements in A(X ) are described by series��

n=1 an #n given above, then we simply have

P$X \ :�

n=1

an #n+=* :�

n=1 \ :n&1

i=1

ai+ #n .

It is well known [1, 10] that, for *=1, P$X defines a Baxter operator ofweight 1 on A(X ). It follows that, for any * # C, P$X defines a Baxteroperator of weight * on A(X ), since it can be easily verified that for anyBaxter operator P of weight 1, the operator *P is a Baxter operator ofweight *. Hence (A(X ), P$X) is in BaxC .

Definition 4.1. Let S(X )0 be the Baxter subalgebra in Bax0C of A(X )

generated by the sequences t(x)=(t (x)1 , ..., t (x)

n , ...), x # X. S(X )0 is called thestandard Baxter algebra on X.

Note that S(X )0 is denoted by S(X ) in Rota's notation. We reserveS(X ) for the free Baxter algebra on X with an identity that will be definedbelow.

Theorem 4.2 (Rota [10]). (S(X )0, P$X) is a free Baxter algebra on Xin the category Bax0

C .

4.2. The Standard Baxter Algebra in General

Given A # AlgC , we now give an alternative construction of a free Baxteralgebra on A in the category BaxC of Baxter algebras with an identity.

For each n # N+ , denote A�n for the tensor power algebra. Denote thedirect limit algebra

A� =� A�n,

where the transition map is given by

A�n [ A� (n+1), x [ x�1A .

Note that the multiplication on A�n here is different from the multiplica-tion on A�n when it is regarded as the C-submodule �n&1

C (A) of �C (A).


To distinguish between the two contexts, we will use the notation �nC (A)

for A� (n+1)��C (A). Let A(A) be the set of sequences with entries in A� .Thus we have

A(A)= `�

n=1

A� #n={ :�

n=1

an#n , an # A� = .

Define addition, multiplication, and scalar multiplication on A(A) com-ponentwise, making A(A) into a A� -algebra, with the sequence (1, 1, ...) asthe identity. Define

P$A=P$A, * : A(A) � A(A)

by

P$A(a1 , a2 , a3 , ...)=*(0, a1 , a1+a2 , a1+a2+a3 , ...).

Then (A(A), P$A) is in BaxC . For each a # A, define t(a)=(t (a)k )k in A(A) by

t (a)k =}

k

i=1

ai \=}�

i=1

ai + , ai={a,1

i=k,i{k.

Definition 4.3. Let S(A) be the Baxter subalgebra in BaxC of A(A)generated by the sequences t(a)=(t (a)

1 , ..., t (a)n , ...), a # A. S(A) is called the

standard Baxter algebra on A.

Since �C (A) is a free Baxter algebra on A, the morphism in AlgC

A � A(A), a [ t(a)

extends uniquely to a morphism in BaxC

8 : �C (A) � A(A).

We will prove in Theorem 4.6 that, when * is not a zero divisor in A� , 8is an isomorphism. Hence (S(A), P$A) is a free Baxter algebra on A in thecategory BaxC . Before proving the theorem, we will first give somenotations and preliminary results.

For k # N+ , denote

F kA(A)=[(ai) # A(A) | ai=0, i�k]

={ :�

n=k+1

an #n } an # A� = .

120 GUO AND KEIGHER

Also denote F 0A(A)=A(A). Clearly each F k A(A) is an ideal of A(A).Define

F kS(A)=F kA(A) & S(A).

Then we have

F kS(A)=[(a i) # S(A) | a i=0, i�k].

F kS(A) are ideals of S(A). Recall from Section 3 that there is a canonical(Baxter) filtration Filk on A(A) and S(A) defined by P$A . We will explainthe relation between them in Lemma 4.10.

Lemma 4.4. For any k # N, we have

(1) A(A) P$A(F kA(A))�F k+1A(A).

(2) 8(Filk �C (A))�F k A(A).

Similar inclusions hold for S(A).

Proof. We only need to verify the inclusions for A(A). The inclusionsfor S(A) follow immediately. By the definition of P$A(F kA(A)) we haveP$A(F kA(A))�F k+1A(A). Since A(A) P$A(F kA(A)) equals the ideal ofA(A) generated by P$A(F kA(A)), we get the first inclusion.

The second inclusion is clear for k=0. By induction on k, for k>0, wehave

8(Filk �C (A))�8(�C (A) PA(Filk&1 �C (A)))

�8(�C (A)) P$A(8(Filk&1 �C (A)))

�A(A) P$A(F k&1 A(A))

�F kA(A). K

Lemma 4.5. For n # N+ and a1 � } } } �an # �nC (A), we have

8(a1 � } } } �an)=*n&1(an � } } } �a1) #n+F nS(A).

Proof. By definition, for a1 # A��C (A),

8(a1)= :�

k=1

t (a1)k #k=a1#1+(1�a1) #2+ } } } .

So the lemma is proved for n=1. Assume that the lemma is proved for n,and consider a1 � } } } �an+1 # �n

C (A). Applying Lemma 4.4, we have


8(a1 �a2 � } } } �an+1)

=8(a1 PA(a2 � } } } �an+1))

=8(a1) 8(PA(an+1 � } } } �a2))

=8(a1) P$A(8(an+1 � } } } �a2))

=\ :�

k=1

t (a1)k #k + P$A(*n&1(an+1 � } } } �a2) #n+a term in F nS(A))

=\ :�

k=1

t (a1)k #k + (*n(an+1 � } } } �a2) #n+1+a term in F n+1S(A))

=*n(an+1 �an � } } } �a1) #n+1+a term in F n+1S(A).

This completes the induction. K

Now we are ready to prove the main theorem of this section.

Theorem 4.6. Assume that * # C is not a zero divisor in A� . Themorphism in BaxC

8 : �C (A) � S(A)

induced by sending a # A to t(a)=(t (a)1 , ..., t (a)

n , ...) is an isomorphism.

Corollary 4.7. When * is not a zero divisor in A� , (S(A), P$A) is a freeBaxter algebra on A in the category BaxC .

Corollary 4.8. Assume that * is not a zero divisor in C. Let X be a set.The morphism in BaxC

8 : �C (X ) � S(X )

induces by sending x # X to t(x)=(t (x)1 , ..., t (x)

n , ...) is an isomorphism. Therestriction of 8 to �C (X )0 is an isomorphism in Bax0

C from �C (X )0 toS(X )0.

Proof. Applying Theorem 4.6 to the case when A=C[X ], we obtain8: �C (X )$S(X ). Since �C (X )0��C (X ) is generated by X in Bax0

C

and S(X )0�S(X )=8(�C (X )) is generated by 8(X ) in Bax0C , the

corollary follows. K

Remarks. (1) The proof of Theorem 4.6, specialized to the setting ofCorollary 4.8, also gives another proof of Theorem 4.2.

(2) The above construction of S(A) for A # AlgC can be modified togive the construction of an internal free Baxter algebra S(A)0 in Bax0

C on

122 GUO AND KEIGHER

A for A # Alg0C . The situation is similar to the construction of shuffle Baxter

algebras not necessarily having an identity, discussed in Section 2.

Proof of Theorem 4.6. Since (�C (A), PA) is a free Baxter algebra on Ain BaxC , the assignment

8 : A � S(A), a [ t(a)=(t (a)1 , ..., t (a)

n , ...)

induces a morphism 8 : �C (A) � S(A) in BaxC . Since S(A) is the Baxtersubalgebra of A(A) generated by A, the morphism 8 is onto. So we onlyneed to verify that 8 is injective.

For any G # �C (X ), we can uniquely write G=�n # N Gn with Gn #�n

C (A). Suppose 8(G)=0: we will show by induction on n # N that Gn=0.For n=0 we have �0

C (A)=A. So G0 is in A. By Lemma 4.4 we have

8 \ :k�1

Gk + # F 1 A(A).

Thus the first component of 8(F ) in A(A) is from

8(G0)=G0 #1+a term in F 1 A(A).

Thus 8(G)=0 implies G0#1=0. Therefore, G0=0.Now assume that Gk=0 for k�n and consider Gn+1 # �n+1

C (A)=A� (n+2). Thus Gn+1 can be expressed as

Gn+1= :k

i=1

a (i )1 � } } } �a (i )

n+2 , k # N+ , a (i)j # A.

Since Gk=0 for k�n, and by Lemma 4.4, for k�n+2,

8(Gk) # 8(�kC (A))

�8(Filk �C (A))

�8(Filn+2 �C (A))

�F n+2S(A),

the only contribution of 8(G) to the coefficient of #n+2 is from 8(Gn+1).By Lemma 4.5, this coefficient is

*n+1 :k

i=1

a (i )n+2 � } } } �a (i )

1 .


Thus 8(F )=0 implies that

*n+1 :k

i=1

a (i )n+2 � } } } �a (i )

1 =0.

Since * is not a zero divisor in A� , we further have

:k

i=1

a (i )n+2 � } } } �a (i )

1 =0

as an element in the tensor power algebra A� (n+2). But this element beingzero or not depends only on the C-module structure of A� (n+2) and theC-module map

A� (n+2) � A� (n+2), an+2 � } } } �a1 [ a1 � } } } �an+2

is an isomorphism. Thus we also have

Gn+1+ :k

i=1

a (i)1 � } } } �a (i )

n+2=0.

This completes the induction. Thus 8 is injective and hence anisomorphism. K

4.3. Completions

We now give an internal construction of free complete Baxter algebrasby showing that the complete free Baxter algebra �� C(A) can be embeddedinto A(A).

Let A(A)$ be the subgroup of A(A) consisting of sequences with finitelymany non-zero entries. Then we have

A(A)$= ��

n=1

A� #n .

Define a filtration on A(A)$ by taking

F kA(A)$=A(X ) & F kA(A)$= ��

n=k+1

A� #n .

By Proposition 3.4, we have

�(A(A)$�Filk A(A)$)$ `k # N+

A�

124 GUO AND KEIGHER

with the addition, multiplication, and scalar multiplication definedcomponentwise. Therefore,

�(A(A)$�Filk A(A)$)$A(A).

From the definition of F k A(A) and F kA(A)$,

A(A)$�Filn A(A)$$A(A)�Filn A(A).

So A(A) is the completion of itself with respect to the filtration F kA(X ).Since F kS(A)=S(A) & F kA(A), we have the injective map of inversesystems

S(A)�F kS(A) � A(A)�F kA(A), k # N+ .

So

�(S(A)�F k S(A))/��(A(A)�F kA(A))$A(A). (7)

We can easily describe the image of �(S(A)�F kS(A)) in A(A). It consistsof sequences (b(n))n , b (n) # A� , that can be expressed as an infinite sum of theform

:�

k=1

(b (n)k )n ,

where (b (n)k )n # S(A) for each k. This means that, for any fixed n # N+ , all

but finitely many b (n)k , k # N+ , are non-zero, and ��

k=1 b (n)k =b(n). We

denote this image by S� (A) with the induced Baxter algebra structure.On the other hand, we also have the Baxter filtration Filk on A(A) and

S(A) (Section 3).

Theorem 4.9. (1) The Baxter algebra A(A) is complete.

(2) Assume that * # C is not a zero divisor in A� . The isomorphism8 : �C(A) � S(A) extends to an isomorphism of complete Baxter algebras

8� : �� C(A) � S� (A).

We first prove a lemma.

Lemma 4.10. For any k # N+ , we have

(1) Filk A(A)=*kF kA(A).

(2) Assume that * # C is not a zero divisor in A� . Filk S(A)=F k S(A).


Proof. (1) We prove that, for any n # N+ ,

P$A(F kA(A))=*F k+1 A(A).

By the definition of P$A : A(A) � A(A), we have P$A(F kA(A))�*F k+1A(A).On the other hand, any element in *F k+1 A(A) is of the form

* :�

i=k+2

ai#i , ai # A� .

Then we have

P$A \ :�

i=k+1

(ai+1&a i) #i +=* :�

i=k+2

ai#i .

Here we take ak+1=0. This proves the equation.When k=1, we have P$A(A(A))=*F 1A(A). Since F 1A(A) is already a

Baxter ideal, we have Fil1 A(A)=*F 1 A(A). Inductively, assuming thatFilk A(A)=*kF kA(A), then we have

P$A(Filk A(A))=P$A(*kF k A(A))=*k+1F k+1A(A).

Since F k+1A(A) is a Baxter ideal of A(A), it is Filk+1 A(A), the Baxterideal generated by P$A(Filk A(A)).

(2) By Lemma 4.4, Lemma 4.5, and Theorem 4.6, we have, for anya # �C(A)

8(a) # Filk S(A) � a # Filk �C(A)

� 8(a) # S(A) & F kA(A)

� 8(a) # F kS(A).

This proves the second equation. K

Proof of Theorem 4.9. From the first equation of Lemma 4.10, we havethe exact sequence of inverse systems

0 � F k A(A�*kA) � A(A)�Filk S(A) � A(A)�F kA(A) � 0.

Clearly � F k A(A�*kA)=0. Thus we have

A� (A)/��(A(A)�F kA(A))=A(A).

This proves the first statement.

126 GUO AND KEIGHER

Next assume that * # C is not a zero divisor in A� . Then by the secondstatement of Lemma 4.10,

S� (A) $ �(S(A)�F kS(A)).

Then by Theorem 4.6 and Eq. (7) we obtain.

�� C(A) $ S� (A)/�A(A). K

REFERENCES

1. G. Baxter, An analytic problem whose solution follows from a simple algebraic identity,Pacific J. Math. 10 (1960), 731�742.

2. P. Cartier, On the structure of free Baxter algebras, Adv. in Math. 9 (1972), 253�265.3. P. M. Cohn, `Ùniversal Algebra,'' Harper and Row, New York, 1965.4. L. Guo and W. Keigher, Baxter algebras and shuffle products, Adv. in Math., in press.5. L. Guo, Properties of free Baxter algebras, Adv. in Math., in press.6. N. Jacobson, ``Basic Algebra, I,'' Freeman, San Francisco, 1980.7. W. Keigher, On the ring of Hurwitz series, Comm. Algebra 25 (1997), 1845�1859.8. S. MacLane, ``Categories for the Working Mathematician,'' Springer-Verlag, New York,

1971.9. H. Matsumura, ``Commutative Ring Theory,'' Cambridge Univ. Press, Cambridge, UK,

1986.10. G. Rota, Baxter algebras and combinatorial identities, I, Bull. Amer. Math. Soc. 5 (1969),

325�329.11. G. Rota, Baxter operators, an introduction, in ``Gian-Carlo Rota on Combinatorics,

Introductory Papers and Commentaries'' (J. P. S. Kung, Ed.), Birkha� user, Boston, 1995.12. G. Rota and D. A. Smith, Fluctuation theory and Baxter algebras, in `Ìstituto Nazionale

di Alta Mathematica, Symposia Mathematica,'' Vol. IX, pp. 179�201, 1972.13. C. Weibel, `Àn Introduction to Homological Algebra,'' Cambridge Univ. Press,

Cambridge, UK, 1994.

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On Free Baxter Algebras: Completions and the Internal Construction